Instability intervals and growth rates for Hill’s equation

Hill’s equation is a real linear second-order ordinary differential equation with a periodic coefficient $f(t)$: \begin{equation} y^{\prime \prime } (t) +\left [ \lambda +\varepsilon f\left (t\right ) \right ] y(t) =0. \tag *{(0.1)} \end{equation} It has unbounded solutions for certain intervals of the real parameter $\lambda$, called instability intervals. Here these intervals, and the growth rate of the unbounded solutions, are determined for $\varepsilon$ small, and also for $\lambda$ large. This is done by constructing a fundamental pair of solutions which are power series in $\varepsilon /\lambda ^{1/2}$, with coefficients that are bounded functions of $\lambda$.